INVARIANT TWINS
constructed, nobody outside of a lunatic asylum has yet suc-
ceeded in visualizing a space of more than three dimensions.
Modern physics is fast teaching some to shed their belief in a
»mysterious 'absolute space' over and above the mathematical
'spaces' - like Euclid's, for example - that were constructed by
geometers to correlate their physical experiences. Geometry
to-day is largely a matter of analysis, but the old terminology
of 'points', 'lines', 'distances', and so on, is helpful in suggesting
interesting things to do with our sets of co-ordinates. But it
does not follow that these particular things are the most useful
that might be done in analysis; it may turn out some day that
all of them are comparative trivialities by more significant
things which we, hidebound in outworn traditions, continue to
do merely because we lack imagination.
If there is any mysterious virtue in talking about situations
which arise in analysis as if we were back with Archimedes
drawing diagrams in the dust, it has yet to be revealed. Pictures
after all may be suitable only for very young children; Lagrange
dispensed entirely with such infantile aids when he composed
bis analytical mechanics. Our propensity to 'geometrize' our
analysis may only be evidence that we have not yet grown up.
Newton himself, it is known, first got his marvellous results
analytically and reclothed them in the demonstrations of an
Apollonius partly because he knew that the multitude -
mathematicians less gifted than himself - would believe a
theorem true only if it were accompanied by a pretty picture and
a skilled Euclidean demonstration, partly because he himself still
lingered by preference in the pre-Cartesian twilight of geometry.
The last of Cayley's great inventions which we have selected
for mention is that of matrices and their algebra in its broad
outline. The subject originated in a memoir of 1858 and grew
directly out of simple observations on the way in which the
transformations (linear) of the theory of algebraic invariants
are combined. Glancing back at what was said on discrirnina.nts
and then: mvariance we note the transformation (the arrow is
<p% JL n
here read 'is replaced by') i/—>--------. Suppose we have two
TX 4~ 3
such transformations,
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